As a result, most MD simulations of oxide glasses have been limited to the sample size on the order of tens of nanometers in size 26 , 27 , Using screened Coulombic interactions, MD simulations of oxide glasses in the submicron scale have been performed 29 , However, contact cracking has not hitherto been observed in MD simulation.
In the present study, crack nucleation under impact indentation is directly captured with full atomic details through MD simulation. Indenter angle and indenter tip radius strongly affect the local stress fields and deformation mechanism. The local stress evolution is carefully examined and the critical local stress conditions for crack nucleation are captured.
Combining the stress state evolution with the fracture criterion, the effects of the indenter angle and tip radius are quantitatively explained.
Our simulations suggest that the commonly used maximum principal fracture criterion might not be sufficient to describe fully the indentation fracture mechanism in brittle oxide glasses.
The simulation setup is illustrated in Fig. Detailed information can be found in the Methods section. As shown in Fig. The tip radius is atomically sharp. The indenter with a larger indenter angle pushes the tensile region away from the indenter tip.
Pile-up is seen under the 60 o indenter and 30 o indenter during loading. For the 90 o indenter, apparent pile-up occurs only after unloading not shown. Under the 30 o indenter, a small crack was nucleated, which can be more clearly seen in the close-up view around the indenter in the upper-right part of Fig. The geometry of the rigid atomic indenter is defined by the angle and tip radius.
Periodic boundary condition PBC applies to the x and y directions. Upper-z direction is a surface. A close-up view around the indenter and the nucleated crack under the 30 o indenter is shown on the right.
Positive negative stress means tensile compressive. The atoms in the indenter are colored gray. The symmetry line is indicated by the white dashed line. The maximum tensile site, i. Note that with a larger indenter angle, the maximum tensile site is farther away from the indenter tip. Hence, we see that a larger indenter tip-radius tends to require a larger indentation depth in order to nucleate a median crack. Equivalently, the fracture load increases with tip-radius as shown in Fig.
A linear relationship between the critical load and tip radius is expected if self-similarity is perfectly obeyed. The slight slope change for larger samples in Fig. More simulations are certainly needed to explain such details. Another detail is that all the median cracks are nucleated sub-surface. With a larger indenter tip radius, the crack nucleation site will be farther away from the tip of the indenter. The close-up views around the indenter and the nucleated crack are shown on the right.
To understand the different crack nucleation behavior caused by various indenter geometries, we first need to understand the local fracture criterion. The locality is defined by a length scale over which the stress state can be regarded as homogeneous. To mimic the stress state in a thick sample and to be consistent with the indentation simulation, we consider the plain strain condition and hence only two degrees of freedom exist, i.
In this way, the fracture stress states can be obtained along the loading paths covering the whole tensile-dominant domain.
The compressive dominant domain is omitted. The upper quarter of space can be obtained by symmetry due to the isotropicity in glasses. Corresponding final deformation morphologies, colored by atomic shear strain 56 , 57 , 58 , are shown on the right. Red crosses collectively represent the crack nucleation stress states for the model glass under plain strain condition. Black circles denote the ultimate yield stress states, i.
Shown on the right, the crack nucleation stress states and the ultimate yield stresses form a locus, which is termed fracture locus. Deviation from the maximum principal stress criterion is highlighted by the black double headed arrows.
In the tensile regime, a maximum deviation of 1 GPa is seen. Blue circles are the Mohr circles for corresponding stress states. In Fig. Therefore, the brittle-to-ductile transition observed here is a true material property and is not caused by having an artificially small sample size.
The stress state corresponding to the brittle fracture stress state is denoted by red crosses. If the glass enters into flow state, the ultimate yield point the outmost stress state is denoted by black circles.
Passing the ultimate yield point, the stress states along different paths become effectively constant, fluctuating around the corresponding fixed stress states, i.
These boundary points form a locus, which we term the fracture criterion or fracture locus. The fracture criterion is the material property that determines the crack nucleation behavior. Although the fracture criterion is obtained under homogeneous loading condition, it is applicable to the local stress state in a non-homogeneous stress field since we can always define a scale within which a uniform stress state can be assumed. The maximum tensile site is illustrated by the white box in Fig.
Note that maximum tensile site is not a physically fixed point in the sample. As the indenter depth increases, it moves further into the glass.
It is clear that such flow stress state is far away from the fracture stress states red crosses and therefore no crack is nucleated. When the indenter angle is reduced to 60 o , the loading path of the maximum tensile site is rotated counter-clockwise in the principal stress domain, approaching to the more tensile direction and towards the fracture stress states.
However, it is still away from the fracture stress states. Once the trajectory touches the fracture line, a crack is nucleated. After the crack nucleation, the maximum tensile site is at the crack tip and the stress state fluctuates around the point 7 GPa, 3 GPa.
Therefore, we can see that the fracture criterion can quantitatively explain the effect of indenter angle in terms of crack nucleation behavior. The fracture criterion and the stress state evolution of the maximum tensile site under different indenters, a for atomically sharp indenters with different angle and b—d for indenters with different tip radius.
The dashed arrows indicate the direction of evolution. For each simulation where a crack is nucleated, the stress state evolution is colored differently after the crack nucleation event to show clearly that the transition state is close to the fracture criterion. For those simulations without crack nucleation, the stress state fluctuates around the flow state. The trajectory eventually reaches the fracture locus, which leads to crack nucleation.
It is interesting to note that once the crack forms, the stress state at the crack tip is nearly the same for all of the simulations, including both loading and unloading. Similar scenarios are observed in all of the simulations with different indenter tip radii, as shown in Fig. For a larger tip radius, a larger indentation depth is required to allow the effect of the indenter angle to take over, a direct result of the particular indenter geometry.
The loading rate is known to have an effect on the indentation fracture behavior 6. It is observed that the fracture locus shrinks or expands when we decrease or increase the strain rate by one order of magnitude. The fracture stress appears to decrease with strain rate in a Arrhenius manner. Therefore, it is important to confirm that the fracture criterion applied to indentation fracture in Fig. Solving this equation, we can estimate the effective strain rate experienced by the crack nucleation site to be 0.
Therefore, it matches the strain rate we use to measure the fracture locus 0. And the indenter depth is on the order of microns. Accordingly, the effective strain rate under the indenter is around 0.
At longer time scale as in quasi-static indentation experiment, the effects of water is expected to further decrease the critical stress for crack nucleation 6 , Also, due to the effect of micro-flaws 34 , crack propagation rather than crack nucleation can be made possible for indenters blunter than the ones studied here. However, we would like to emphasize that the presence of flaws will not affect the applicability of our local fracture locus.
It is also very interesting to note that in experiment at macroscopic scale, median crack can be caused by shear localization and then shear faults under blunt indenters 6 , According to our previous study, shear band to crack transition has a length scale beyond nano-scale 24 even in metallic glasses which are much more prone to form a shear band.
Shear band in oxide glasses are much less ready to occur due to less pronounced shear softening. Therefore shear banding and shear band to crack transition in oxide glasses might require a much larger sample size that is beyond nanoscale or submicron scale. As a result, shear faults or shear bands have not been observed in oxide glasses in MD simulations 27 , 28 , Further studies on these extrinsic factors are underway. Fracture locus shrinks with decreasing strain rate.
The strain rate is shown in a log scale. One important implication of this study is that the widely used maximum principal stress fracture criterion for brittle materials might not be sufficient for indentation fracture in glasses. The influence of shear stress can be seen in the equivalent fracture locus in Fig.
Shear dominant stress state is typical at the maximum tensile site under the indenter as can be seen in Fig. Due to the presence of the shear dominant stress state, non-negligible shear or plastic deformation occurs under the indenter. A noticeable plastic zone and even shear bands have been observed in oxide glasses in experiments 6 , 22 , Plastic flow induced pile-up is also observed in current and previous MD simulations 27 , 30 and in nano-indentation experiments 8 , 9 , 35 , 36 , This is in agreement with previous simulations 24 and experimental observations 6 that shear faults or shear flow can dramatically decrease the fracture stress.
Our current MD simulations suggest that to achieve accurate prediction of indentation fracture, both in-plane principal stresses are required under plane strain condition and all three principal stresses might be required under general stress state. In the shear or compressive dominant stress state, local ductility should also be considered even for brittle glasses. In summary, based on MD simulations we have developed a crack nucleation map in the principal stress domain to quantitatively explain the crack nucleation behavior under impact indentations with different indenter angle and indenter tip radius.
The primary idea of the map is to decompose the complicated indenation fracture problem into two components. The first component is the local fracture criterion, which is an intrinsic material property that is mostly controlled by the glass composition and loading rate. The second component is the stress evolution of the maximum tensile site under indentation, which is controlled mostly by extrinsic parameters such as the indenter geometries.
Due to the large shear or compressive stress component under indentation, the commonly held maximum principal stress fracture criterion might not be sufficient even for brittle glasses under indentation.
The force field has been used in previous MD studies to simulate the mechanical properties of oxide glasses 40 , 41 , The final dimension of the sample was 8.
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